Energy methods and H-convergence

For nonsymmetric matrices it is convenient to use two quadratic bounds: A is coercive with constant α, and its inverse is coercive with constant 1/β. This class controls the matrix and is stable under H-convergence. A simpler original assumption ξ·Aξ≥λ|ξ|² and |Aξ|≤Λ|ξ| places A in this class with α=λ and β=Λ²/λ. One should not automatically promise that the same original operator-norm bound Λ survives for a nonsymmetric effective coefficient.

Given a coefficient Aₙ, let uₙ solve its weak Dirichlet equation for a fixed source. H-convergence to A₀ requires uₙ to converge weakly in H₀¹ to the A₀ solution and Aₙ∇uₙ to converge weakly in L² to A₀∇u₀, for every such source. The flux condition is part of the definition. Uniform ellipticity gives compactness in this sense along a subsequence, but the limit may depend on position and on the subsequence if no additional structure identifies it.

Testing the equation with its own solution gives the total quadratic quantity ∫Aₙ∇uₙ·∇uₙ=⟨f,uₙ⟩. The right side converges under weak H₀¹ convergence because f is fixed in its dual. More general local product passages use compensated compactness: gradients have zero curl, while flux divergences are controlled by the equation. This structure can compensate for the absence of ordinary strong convergence. It is not a general rule that two weakly convergent L² sequences have convergent products.

For symmetric coefficients, the Dirichlet solutions minimize quadratic energies, and classical G-convergence can be defined by their weak convergence for every source. In that symmetric setting it agrees with the corresponding H-convergence, and localized energy Γ-convergence provides a variational characterization. Here the term G-convergence is reserved for that symmetric conductivity framework. Extending a solution-only definition to nonsymmetric matrices without attending to flux can hide information that the energy's symmetric part cannot recover.

A constant skew-symmetric matrix K satisfies div(K∇u)=0 in distributions, because mixed derivatives commute and its paired coefficients cancel. Consequently A=I+K and I give the same scalar Dirichlet equation for every source, while their constitutive fluxes differ by K∇u. This does not violate uniqueness of the weak solution. It shows that the solution operator alone does not specify every nonsymmetric flux representation of that differential operator.

The H-limit retains the chosen flux law, so two constant sequences with those respective matrices have different H-limits despite identical potential solutions. In symmetric problems the quadratic energy does identify the matrix through polarization, removing this particular ambiguity. These distinctions matter when the observable is a current, stress, or transport direction rather than only a scalar potential. A convergence claim should say which field is being predicted and which coefficient class its theorem covers.